{
 "artifacts": null,
 "category": "training",
 "description": "Replace ordinary adversarial training over a fixed perturbation set with adaptive robust training in which the admissible perturbations depend on the current network state. Train on a small active set of hard scenarios, then search for a newly admissible scenario with larger loss or constraint violation and add it only when needed. This should reduce redundant adversarial examples while targeting worst-case regions induced by the current model.",
 "formulas_latex": [
  "$$\\min_{z\\in\\mathcal{Z}} f(z)\\quad\\text{s.t.}\\quad g(z,v)\\leq0\\qquad\\forall v\\in\\mathcal{V}(z).$$",
  "$$\\mathcal{V}(z)=\\left\\{v\\in\\bar{\\mathcal{V}}\\;\\middle|\\;h(z,v)\\geq0,\\ r(z,v)=0\\right\\}.$$",
  "$$\\Phi_\\tau(\\theta,v)=g(\\theta,v)-\\lambda_h\\tau\\log\\!\\sum_{i=1}^{n_h}e^{-h_i(\\theta,v)/\\tau}-\\lambda_r\\|r(\\theta,v)\\|_2^2.$$",
  "$$F_A(\\theta)=f(\\theta)+\\beta\\tau_A\\log\\!\\sum_{v_k\\in A}\\exp\\!\\left(g(\\theta,v_k)/\\tau_A\\right).$$"
 ],
 "id": 3011,
 "implementation": "Integrate the method at the training-loss level for a classifier or policy network. Let $\\theta$ denote network parameters, $x$ a minibatch input, and $v$ encode a perturbation or environment scenario. Define $g(\\theta,v)=\\ell(f_\\theta(x,v),y)-\\rho$. Define $h(\\theta,v)$ and $r(\\theta,v)$ to represent model-dependent admissibility; for example, use a perturbation radius depending on predictive entropy, or require a representation-preservation equality $r(\\theta,v)=\\|\\phi_\\theta(x+v)-\\phi_\\theta(x)\\|_2-c=0$. Maintain an active set $A$ initialized with the clean scenario and several random admissible scenarios. For each outer iteration, perform $T_A$ gradient updates on $F_A(\\theta)$ using the displayed log-sum-exp formula. Then freeze $\\theta$ and run an inner oracle from the worst active scenario plus random restarts: maximize $\\Phi_\\tau(\\theta,v)$ over bounded $v\\in\\bar{\\mathcal V}$ using projected Adam or projected gradient ascent. Gradually decrease $\\tau$ and increase $\\lambda_h,\\lambda_r$; accept a candidate only if numerical feasibility satisfies $\\min_i h_i(\\theta,v)\\geq-\\epsilon_h$ and $\\|r(\\theta,v)\\|_2\\leq\\epsilon_r$. If its loss exceeds the current active maximum by more than $\\epsilon_g$, append it to $A$; otherwise end the cutting-plane round. Stop gradients through $\\theta$ during oracle search. The formulas, active-set logic, and separation criterion come from the paper; the penalty relaxation and scenario-loss definitions are the neural-network adaptation. First test on CIFAR-10 with ResNet-18 against standard PGD adversarial training under the same total inner-gradient budget. Use a radius depending on predictive entropy, evaluate with an independent constrained multi-start oracle, and measure robust accuracy, clean accuracy, active-set size, oracle calls, and total FLOPs. A useful signal is lower worst-case loss with fewer retained scenarios or faster loss descent at equal attack-gradient cost; failure is unbounded active-set growth, poor feasibility, or no gain over PGD.",
 "math_summary": "The paper formulates a generalized semi-infinite program as $\\min_{z\\in\\mathcal{Z}} f(z)$ subject to $g(z,v)\\leq0$ for every $v\\in\\mathcal{V}(z)$, where $z$ is the finite-dimensional decision, $v$ is an index or uncertainty variable, $f$ is the objective, and $g$ is a robust constraint. The decision-dependent uncertainty set is $\\mathcal{V}(z)=\\{v\\in\\bar{\\mathcal{V}}\\mid h(z,v)\\geq0,\\ r(z,v)=0\\}$, where $\\bar{\\mathcal{V}}$ is a bounded fixed superset, $h$ contains inequality feasibility functions, and $r$ contains equality feasibility functions. Adapt $z$ to neural-network parameters $\\theta$, let $v$ be an input perturbation or environment scenario, and define $g(\\theta,v)=\\ell(\\theta;v)-\\rho$, where $\\ell$ is scenario loss and $\\rho$ is a target loss. The inner oracle searches for a feasible $v$ maximizing $g$. To make the oracle differentiable, use the penalty $\\Phi_\\tau(\\theta,v)=g(\\theta,v)-\\lambda_h\\tau\\log\\sum_i\\exp(-h_i(\\theta,v)/\\tau)-\\lambda_r\\|r(\\theta,v)\\|_2^2$, with temperature $\\tau\u003e0$ and penalty weights $\\lambda_h,\\lambda_r\u003e0$. For an active scenario set $A$, optimize the smooth robust surrogate $F_A(\\theta)=f(\\theta)+\\beta\\tau_A\\log\\sum_{v_k\\in A}\\exp(g(\\theta,v_k)/\\tau_A)$, where $\\beta$ is robust-loss weight and $\\tau_A$ controls approximation to the maximum. The key property is finite active-set optimization combined with separation: the oracle either finds a violating admissible scenario or certifies that the current discretization is approximately robust.",
 "math_tags": [
  "optimization",
  "logic",
  "numerical-analysis",
  "dynamical-systems"
 ],
 "ml_areas": [
  "training",
  "regularization",
  "optimizer",
  "rl"
 ],
 "paper": {
  "arxiv_id": "2609.01538",
  "arxiv_url": "https://arxiv.org/abs/2609.01538",
  "summary_what_math_gives_to_ml": "The paper gives a constructive treatment of universal constraints whose uncertainty set changes with the decision variable, using adaptive discretization rather than repeatedly solving the full robust problem globally. The transferable asset is an active-set loop: optimize against a finite set of difficult scenarios, then search for a newly admissible scenario that violates the current solution and add it only when necessary. In neural networks this enables decision-dependent robustness, where perturbation or environment admissibility depends on the current representation, prediction, policy, or parameters. The most practical experiment is a cutting-plane adversarial-training method with differentiable penalties for the inner feasibility conditions and an independent constrained oracle for evaluation.",
  "title": "Generalized Semi-Infinite Programming for Robust Optimal Control with Decision-Dependent Uncertainty",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 6
 },
 "solves": [
  "accuracy",
  "stability",
  "sample-efficiency"
 ],
 "title": "Decision-Dependent Robustness Cutting Planes",
 "url": "https://synthcore.org/idea/3011/decision-dependent-robustness-cutting-planes",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}